Related papers: The Brezis-Nirenberg problem in 4D

The one-dimensional Schr\"{o}dinger equation for a class of potentials $V(|x|)$ which vanish at infinity and present dominant singularity at the origin in the form $\alpha /|x|^{\beta}$ ($0<\beta \leq 2$) is investigated. The Hermiticity of…

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{\theta(p-1)}}\right) = h(u)f…

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\…

In this paper we consider the model semilinear Neumann system $$\left\{ \begin{array}{lll} -\Delta u+a(x)u=\lambda c(x) F_u(u,v)& {\rm in} & \Omega,\\ -\Delta v+b(x)v=\lambda c(x) F_v(u,v)& {\rm in} & \Omega,\\ \frac{\partial u}{\partial…

We prove an existence result for Robin boundary value problems modeled on \[ \begin{cases} \Delta u + |\nabla u|^2 + \lambda f(x) = 0 & \text{in } \Omega \\ \frac{\partial u}{\partial \nu} + \beta u = 0 & \text{on } \partial\Omega…

We consider the boundary value problem \begin{equation*} - \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega) \eqno{(P_{\lambda})} \end{equation*} where $\Omega \subset \R^N, N \geq 3$ is…

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u & = & v^q+\a g & \text{in }\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\l f &\text{in }\Omega , \\…

We consider the boundary value problem $-\Delta_p u_\lambda -\Delta_q u_\lambda =\lambda g(x) u_\lambda^{-\beta}$ in $\Omega$ , $u_\lambda=0$ on $\partial \Omega$ with $u_\lambda>0$ in $\Omega.$ We assume $\Omega$ is a bounded open set in…

We consider the semilinear problem \[ \Delta u = \lambda_+ \left(-\log u^+\right) 1_{\{u > 0\}} - \lambda_- \left(-\log u^- \right) 1_{\{u < 0\}} \qquad \hbox{ in } B_1, \] where $B_1$ is the unit ball in $\mathbb{R}^n$ and assume…

We study the fractional laplacian problem (-\Delta)^s u &=& u^p -\epsilon u^q \quad\text{in }\quad \Omega, u &\in& H^s(\Omega)\cap L^{q+1}(\Omega),u &>&0 \quad\text{in }\quad \Omega, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminus\Omega,…

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight…

In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $\beta : \mathbb{R} \to 2 ^{…

This work deals with the system $(-\Delta)^m u= a(x) v^p$, $(-\Delta)^m v=b(x) u^q$ with Dirichlet boundary condition in a domain $\Omega\subset\RR^n$, where $\Omega$ is a ball if $n\ge 3$ or a smooth perturbation of a ball when $n=2$. We…

In this article we address the regularity of stable solutions to semilinear elliptic equations $-\Delta u = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $\Omega \subset \mathbb{R}^n$ and…

In this survey we provide an overview of nonlinear elliptic homogeneous boundary value problems featuring singular zero-order terms with respect to the unknown variable whose prototype equation is $$ -\Delta u = {u^{-\gamma}} \ \text{in}\…

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and…

Motivated mainly by the localization over an open bounded set $\Omega$ of $\mathbb R^n$ of solutions of the Schr\"odinger equations, we consider the Schr\"odinger equation over $\Omega$ with a very singular potential $V(x) \ge C d (x,…

We go further in the investigation of the Robin problem $(P_{\alpha})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $u\geq0$ in $\Omega$, $\partial_{\nu}u=\alpha u$ on $\partial \Omega$; on a bounded domain $\Omega\subset\mathbb{R}^{N}$, with $a$…

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving the eigenvalue problem $Lu=\lambda u$ for an elliptic partial differential…

he equation $-\Delta u = \lambda e^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when $\lambda = 2(N-2)$. If $N\ge 4$ we show that…